Define the loss function \(\ell _\theta (x,y) = \frac{1}{2} \lbrace 1 - y \, \text{sign}(x-\theta )\rbrace \)  and the corresponding risk (expected loss) \(R(\vartheta ) = P \ell _\vartheta \) . As [1]}, showed, the MCID is the minimizer of \(R\) , i.e., \(\theta ^\star = \arg \min _\theta R(\theta )\) . So the goal is to construct an empirical version of the risk function, and then a sort of posterior distribution that will concentrate around values that make the empirical risk small. For the empirical risk, let
\( R_n(\theta ) = \frac{1}{n} \sum _{i=1}^n \ell _\theta (X_i, Y_i). \) 
